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Real partitions of measure spaces. (English. Russian original) Zbl 0832.28012
Sib. Math. J. 35, No. 1, 189-191 (1994); translation from Sib. Mat. Zh. 35, No. 1, 207-209 (1994).
In this short note the author defines the notion of real partition and with its help existence is proved for measure-preserving mappings of several probability spaces onto the unit interval with the Lebesgue measure.
A real partition of a probability space \((X, {\mathcal A}, \mu)\) is defined to be a partition \((D_t)_{t \in [0,1]}\) of the space, such that \(\mu (A^-_t) = \mu (A^+_t) = t\) for all \(t \in [0,1]\), where \(A^-_t = \cup_{s < t} D_s\) and \(A^+_t = \cup_{s \leq t} D_s\).
The main theorem shows that: For every complete nonatomic probability space which includes a Cantor set, there exists a measure-preserving epimorphism of the space onto the unit interval with the Lebesgue measure.
28A99 Classical measure theory
28D05 Measure-preserving transformations
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